Setup

This notebook requires:

  1. LD filtered genotype
  2. Gene expression matrix

Load data 

#collapse-hide
dosagefile = '/cbscratch/sbanerj/gtex_pca/gtex_v8_filtered.dosage.raw'
dosage_numpy_file = '/cbscratch/sbanerj/gtex_pca/gtex_dosage.npy'
expression_file = '/scratch/sbanerj/trans-eqtl/input/gtex_v8/expression/gtex_ms_raw_std_protein_coding_lncRNA.txt'

Load Python libraries 

#collapse-hide
import numpy as np
import pandas as pd
from sklearn.decomposition import PCA
from scipy import stats
import os
from scipy.cluster import hierarchy as hc

import matplotlib.pyplot as plt
import matplotlib
from mpl_toolkits.axes_grid1 import make_axes_locatable
from utils import mpl_stylesheet
mpl_stylesheet.banskt_presentation(fontfamily = 'latex-clearsans', fontsize = 18, colors = 'banskt', dpi = 300)

Read input data 

#collapse-hide
def read_gtex(filename): # returns N x G gene expression
    expr_list = list()
    donor_list = list()
    gene_list = list()
    with open(filename) as mfile:
        donor_list = mfile.readline().strip().split("\t")[1:]
        for line in mfile:
            linesplit = line.strip().split("\t")
            gene = linesplit[0].strip()
            gene_list.append(gene)
            expr = np.array([float(x) for x in linesplit[1:]])
            expr_list.append(expr)
    expr = np.transpose(np.array(expr_list))
    return expr, donor_list, gene_list

if not os.path.isfile(dosage_numpy_file):
    dosage = np.loadtxt(dosagefile, delimiter=' ', skiprows=1, usecols=range(6, 97612))
    np.save(dosage_numpy_file, dosage)
else:
    dosage = np.load(dosage_numpy_file)
gtsamples = list()
with open (dosagefile, 'r') as infile:
    next(infile)
    for line in infile:
        gtsamples.append(line.strip().split()[1])

gx, gxsamples, _ = read_gtex(expression_file)
sampleidx = [gtsamples.index(x) for x in gxsamples] # assumes all expression samples have genotype
dreduce = dosage[sampleidx, :]
gt = dreduce - np.mean(dreduce, axis = 0).reshape(1, -1)

print(f'{len(sampleidx)} samples, {gx.shape[1]} genes, {gt.shape[1]} SNPs.')
print(f'Centered and normalized genotype and expression. Samples in same order as `gxsamples`')

706 samples, 13236 genes, 97606 SNPs.
Centered and normalized genotype and expression. Samples in same order as `gxsamples`

Core functionality modules

  1. Calculate PCA, distance matrix and KNN.
  2. Map distance matrix from one sample space to another.
  3. Remove first N principal components from any matrix X

#collapse-hide
def get_pca(x, K):
    pca = PCA(n_components=K)
    pca.fit(x) # requires N x P (n_samples, n_features)
    x_pca = pca.transform(x)
    return x_pca

def get_distance(a, b):
    return np.linalg.norm(a - b)

def distance_matrix(x_pca):
    nsample = x_pca.shape[0]
    distance_matrix = np.zeros((nsample, nsample))
    for i in range(nsample):
        for j in range(i+1, nsample):
            dist = get_distance(x_pca[i,:], x_pca[j,:])
            distance_matrix[i, j] = dist
            distance_matrix[j, i] = dist
    return distance_matrix

def map_distance_matrix(dm, samples, target_samples):
    N = len(target_samples)
    newdm = np.zeros((N, N))
    newdm[:] = np.nan
    for i in range(N):
        if target_samples[i] in samples:
            newdm[i, i] = 0 # diagonal is always zero
            iold = samples.index(target_samples[i])
            for j in range(i+1, N):
                if target_samples[j] in samples:
                    jold = samples.index(target_samples[j])
                    newdm[i, j] = dm[iold, jold]
                    newdm[j, i] = dm[jold, iold]
    return newdm

def knn(gx, gt, dm, K):
    assert (gx.shape[0] == gt.shape[0])
    N = gx.shape[0]
    gx_knn = np.zeros_like(gx)
    gt_knn = np.zeros_like(gt)

    for i in range(N):
        #neighbors = np.argsort(distance_matrix[i, :kneighbor + 1])
        neighbors = np.argsort(dm[i, :])[:K + 1][1:]
        gx_knn[i, :] = gx[i, :] - np.mean(gx[neighbors, :], axis = 0)
        gt_knn[:, i] = gt[:, i] - np.mean(gt[:, neighbors[1:]], axis = 1)
        
    return gx_knn, gt_knn

def remove_nfirst_pcs(X, n=1):    
    mu = np.mean(X, axis = 0)
    Xnorm = X - mu
    U, S, Vt = np.linalg.svd(Xnorm, full_matrices=False)
    Xhat = U[:, n:] @ np.diag(S[n:]) @ Vt[n:, :] 
    Xhat += mu
    return Xhat

def remove_nfirst_pcs_sklearn(X, n = 1):
    '''
    This is the same as above but using the PCA module of sklearn
    '''
    nsamples = X.shape[0]
    nfeatures = X.shape[1]
    ncomp = min(nsamples, nfeatures)
    pca = PCA(n_components = ncomp)
    pca.fit(X)
    Xpcs = pca.transform(X)
    Xeig  = pca.components_
    mu = np.mean(X, axis=0)
    Xhat = np.dot(Xpcs[:, n:], Xeig[n:, :])
    Xhat += mu
    return Xhat


#collapse-hide
def plot_distance_matrices(dmA, dmB, norms = None):
    '''
    provide norms, if required, as norms = (norm1, norm2)
    where,
    norm1 = matplotlib.colors.DivergingNorm(vmin=10., vcenter=90., vmax=170.)
    norm2 = matplotlib.colors.DivergingNorm(vmin=0., vcenter=90., vmax=300.)
    '''
    
    fig = plt.figure(figsize = (12, 6))
    ax1 = fig.add_subplot(121)
    ax2 = fig.add_subplot(122)
    
    # the zero distance between the same samples
    # is bad for the color scale.
    dmA[np.diag_indices(dmA.shape[0])] = np.nan
    dmB[np.diag_indices(dmB.shape[0])] = np.nan

    cmap1 = plt.get_cmap("YlOrRd")
    cmap1.set_bad('w')

    cmap2 = plt.get_cmap("YlGnBu")
    cmap2.set_bad('w')

    if norms is not None:
        norm1 = norms[0]
        norm2 = norms[1]
        im1 = ax1.imshow(dmA, cmap = cmap1, norm = norm1, interpolation='nearest')
        im2 = ax2.imshow(dmB, cmap = cmap2, norm = norm2, interpolation='nearest')
    else:
        im1 = ax1.imshow(dmA, cmap = cmap1, interpolation='nearest')
        im2 = ax2.imshow(dmB, cmap = cmap2, interpolation='nearest')        


    divider = make_axes_locatable(ax1)
    cax = divider.append_axes("right", size="5%", pad=0.2)
    cbar = plt.colorbar(im1, cax=cax, fraction = 0.1)

    divider = make_axes_locatable(ax2)
    cax = divider.append_axes("right", size="5%", pad=0.2)
    cbar = plt.colorbar(im2, cax=cax, fraction = 0.1)

    ax1.set_title("Genotype space", pad = 20)
    ax2.set_title("Expression space", pad = 20)
    
    plt.tight_layout()
    return fig

Calculation 

#collapse-show
# Before KNN
dm_gt = distance_matrix(get_pca(gt, 20))
dm_gx = distance_matrix(get_pca(gx, 30))

# Expression KNN
K = 30
gx_knn, gt_knn = knn(gx, gt, dm_gx, K)
dm_gt_knn = distance_matrix(get_pca(gt_knn, 40))
dm_gx_knn = distance_matrix(get_pca(gx_knn, gx_knn.shape[0]))

# Double KNN
K1 = 10
K2 = 30
gx_knn1, gt_knn1 = knn(gx, gt, dm_gt, K1)
dm_gx1 = distance_matrix(get_pca(gx_knn1, gx_knn1.shape[0]))
gx_knn2, gt_knn2 = knn(gx_knn1, gt_knn1, dm_gx1, K2)

dm_gt_knn2 = distance_matrix(get_pca(gt_knn2, 40))
dm_gx_knn2 = distance_matrix(get_pca(gx_knn2, gx_knn2.shape[0]))

Option 1. Order samples in genotype space 

#collapse-hide
o1 = hc.leaves_list(hc.linkage(dm_gt, method = 'centroid'))

/usr/users/sbanerj/miniconda3/envs/py36/lib/python3.7/site-packages/ipykernel_launcher.py:2: ClusterWarning: scipy.cluster: The symmetric non-negative hollow observation matrix looks suspiciously like an uncondensed distance matrix
  


#collapse-hide
norm1 = matplotlib.colors.DivergingNorm(vmin=10., vcenter=90., vmax=170.)
norm2 = matplotlib.colors.DivergingNorm(vmin=0., vcenter=75., vmax=300.)
norms = (norm1, norm2)


#collapse-show
mgt = dm_gt[o1, :][:, o1]
mgx = dm_gx[o1, :][:, o1]
fig = plot_distance_matrices(mgt, mgx, norms = norms)
fig.suptitle("Distance between samples before KNN")
plt.show()


#collapse-show
mgt = dm_gt_knn[o1, :][:, o1]
mgx = dm_gx_knn[o1, :][:, o1]
fig = plot_distance_matrices(mgt, mgx, norms = norms)
fig.suptitle("Distance between samples after KNN")
plt.show()


#collapse-show
mgt = dm_gt_knn2[o1, :][:, o1]
mgx = dm_gx_knn2[o1, :][:, o1]
fig = plot_distance_matrices(mgt, mgx, norms = norms)
fig.suptitle("Distance between samples after double KNN")
plt.show()

Option 2. Order samples in expression space 

#collapse-hide
o2 = hc.leaves_list(hc.linkage(dm_gx, method='centroid'))

/usr/users/sbanerj/miniconda3/envs/py36/lib/python3.7/site-packages/ipykernel_launcher.py:2: ClusterWarning: scipy.cluster: The symmetric non-negative hollow observation matrix looks suspiciously like an uncondensed distance matrix
  


#collapse-show
mgt = dm_gt[o2, :][:, o2]
mgx = dm_gx[o2, :][:, o2]
fig = plot_distance_matrices(mgt, mgx, norms = norms)
fig.suptitle("Distance between samples before KNN")
plt.show()


#collapse-show
mgt = dm_gt_knn[o2, :][:, o2]
mgx = dm_gx_knn[o2, :][:, o2]
fig = plot_distance_matrices(mgt, mgx, norms = norms)
fig.suptitle("Distance between samples after KNN")
plt.show()


#collapse-show
mgt = dm_gt_knn2[o2, :][:, o2]
mgx = dm_gx_knn2[o2, :][:, o2]
fig = plot_distance_matrices(mgt, mgx, norms = norms)
fig.suptitle("Distance between samples after double KNN")
plt.show()

Difference between KNN and double-KNN 

#collapse-show
mgt = dm_gt_knn2[o1, :][:, o1] - dm_gt_knn[o1, :][:, o1]
mgx = dm_gx_knn2[o1, :][:, o1] - dm_gx_knn[o1, :][:, o1]
fig = plot_distance_matrices(mgt, mgx)
plt.show()

Calculation with modified distance matrix

Due to strong patterns in the distance matrix, we hypothesized that it could be due to technical noise among the samples. Hence, they are preventing finding the correct neighbors.

Below, I calculated the distance matrix and subtracted the first 2 principal components from the distance matrix. The neighbors for KNN correction were then calculated from the modified distance matrix.

Variables have been reassigned. Poor coding. Be careful!

#collapse-show
# Before KNN
dm_gt = distance_matrix(get_pca(gt, 20))
dm_gx = distance_matrix(get_pca(gx, 30))
dm_gx_corr = remove_nfirst_pcs(dm_gx, n=2)

# Expression KNN
K = 30
gx_knn, gt_knn = knn(gx, gt, dm_gx_corr, K)
dm_gt_knn = distance_matrix(get_pca(gt_knn, 40))
dm_gx_knn = distance_matrix(get_pca(gx_knn, gx_knn.shape[0]))

# Double KNN
K1 = 10
K2 = 30
gx_knn1, gt_knn1 = knn(gx, gt, dm_gt, K1)
dm_gx1 = distance_matrix(get_pca(gx_knn1, gx_knn1.shape[0]))
dm_gx1_corr = remove_nfirst_pcs(dm_gx1, n=2)
gx_knn2, gt_knn2 = knn(gx_knn1, gt_knn1, dm_gx1_corr, K2)

dm_gt_knn2 = distance_matrix(get_pca(gt_knn2, 40))
dm_gx_knn2 = distance_matrix(get_pca(gx_knn2, gx_knn2.shape[0]))

Order samples in expression space

With the modified distance matrix, ordering in expression space and genotype space becomes similar. Probably, we removed other covariates, and population substructure becomes the most important confounder.

#collapse-hide
o2 = hc.leaves_list(hc.linkage(dm_gx_corr, method = 'centroid'))


#collapse-hide
norm1 = matplotlib.colors.DivergingNorm(vmin=10., vcenter=90., vmax=170.)
norm2 = matplotlib.colors.DivergingNorm(vmin=0., vcenter=80., vmax=300.)
norms = (norm1, norm2)


#collapse-hide
mgt = dm_gt[o2, :][:, o2]
mgx = dm_gx_corr[o2, :][:, o2]
fig = plot_distance_matrices(mgt, mgx, norms = norms)
fig.suptitle("Modified distance matrix before KNN")
plt.show()


#collapse-hide
mgt = dm_gt_knn[o2, :][:, o2]
mgx = dm_gx_knn[o2, :][:, o2]
fig = plot_distance_matrices(mgt, mgx, norms = norms)
fig.suptitle("Original distance matrix after KNN")
plt.show()

mgt = dm_gt_knn2[o2, :][:, o2]
mgx = dm_gx_knn2[o2, :][:, o2]
fig = plot_distance_matrices(mgt, mgx, norms = norms)
fig.suptitle("Original distance matrix after double KNN")
plt.show()